versprühen
to spray
To distribute a liquid in very fine droplets in the air, for example with a spray can.
Der Gärtner versprüht ein Mittel gegen Schädlinge auf den Rosen.
The gardener sprays a pesticide on the roses.

to spray / to radiate
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to spray
To distribute a liquid in very fine droplets in the air, for example with a spray can.
Der Gärtner versprüht ein Mittel gegen Schädlinge auf den Rosen.
The gardener sprays a pesticide on the roses.

'Versprühen' (to spray / to radiate) is an inseparable verb. The prefix 'ver-' stays firmly attached to the verb 'sprühen' (to spray) in all sentence forms.
Sie versprühte etwas Parfüm auf ihr Handgelenk.
She sprayed some perfume on her wrist.
You can recognize the meaning from the object: If it is a liquid like 'Wasser' (water) or 'Parfüm' (perfume), the literal meaning is intended. If it is something abstract like 'Charme' (charm) or 'Optimismus' (optimism), the figurative meaning is intended.
Er versprüht am Morgen Wasser auf seine Pflanzen.
He sprays water on his plants in the morning.
In the figurative meaning, 'versprühen' (to radiate) is often combined with positive, abstract nouns. Typical examples are: 'Charme versprühen' (to radiate charm), 'gute Laune versprühen' (to radiate good humor), 'Energie versprühen' (to radiate energy), or 'Lebensfreude versprühen' (to radiate joy of life).
Trotz der Schwierigkeiten versprühte sie immer Optimismus.
Despite the difficulties, she always radiated optimism.
to spread
'verbreiten' and 'versprühen' can both describe something making itself noticeable in the surroundings. With moods or atmosphere, they are similar: a person or a place can spread or radiate joy, calm, or good cheer. 'verbreiten' tends to say that something spreads through a space or among people. 'versprühen' sounds more lively and personal, as if someone is radiating energy or charm. In addition, 'versprühen' is used very concretely for liquid in fine droplets. That is why the words can be confused, but 'Parfüm versprühen' (to spray perfume) is concrete, while 'gute Stimmung verbreiten' (to spread a good mood) sounds more neutral.

Sofia versprüht bei der Feier so viel gute Laune, dass alle mitlachen.
Sofia radiates so much good cheer at the party that everyone laughs along.
Die Kerzen auf dem Tisch verbreiten eine ruhige Stimmung im Wohnzimmer.
The candles on the table spread a calm atmosphere in the living room.